Evolutionary algorithms are popular algorithms for multi-objective optimisation (also called Pareto optimisation) as they use a population to store trade-offs between different objectives. Despite their popularity, the theoretical foundation of multi-objective evolutionary optimisation (EMO) is still in its early development. Fundamental questions such as the benefits of the crossover operator are still not fully understood. We provide a theoretical analysis of the well-known EMO algorithms GSEMO and NSGA-II to showcase the possible advantages of crossover: we propose classes of “royal road” functions on which these algorithms cover the whole Pareto front in expected polynomial time if crossover is being used. But when disabling crossover, they require exponential time in expectation to cover the Pareto front. The latter even holds for a large class of black-box algorithms using any elitist selection and any unbiased mutation operator. Moreover, even the expected time to create a single Pareto-optimal search point is exponential. We provide two different function classes, one tailored for one-point crossover and another one tailored for uniform crossover, and we show that some immune-inspired hypermutations cannot avoid exponential optimisation times. Our work shows the first example of an exponential performance gap through the use of crossover for the widely used NSGA-II algorithm and contributes to a deeper understanding of its limitations and capabilities.

中文翻译：

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交叉可以保证进化多目标优化的指数加速

进化算法是多目标优化（也称为帕累托优化）的流行算法，因为它们使用群体来存储不同目标之间的权衡。尽管多目标进化优化（EMO）很受欢迎，但其理论基础仍处于早期发展阶段。诸如交叉算子的好处之类的基本问题仍未完全理解。我们对著名的 EMO 算法 GSEMO 和 NSGA-II 进行了理论分析，以展示交叉的可能优势：我们提出了“王道”函数类别，如果交叉是，这些算法在预期多项式时间内覆盖整个帕累托前沿。正在使用。但当禁用交叉时，他们需要指数时间来覆盖帕累托前沿。后者甚至适用于使用任何精英选择和任何无偏变异算子的一大类黑盒算法。此外，即使创建单个帕累托最优搜索点的预期时间也是指数级的。我们提供了两种不同的函数类，一种是为单点交叉定制的，另一种是为均匀交叉定制的，并且我们表明，一些免疫启发的超突变无法避免指数优化时间。我们的工作展示了通过对广泛使用的 NSGA-II 算法使用交叉来实现指数性能差距的第一个示例，并有助于更深入地了解其局限性和功能。